Aranda Orna, Diego
Elduque Palomo, Alberto Carlos (dir.)
Universidad de Zaragoza,
2017
Resumen: The main goal of this thesis is to shed some light on the problem of classification of fine gradings on some simple Lie algebras. (Note that the classification of equivalence classes of fine gradings is already known for simple Lie algebras over an algebraically closed field of characteristic zero, but the problem is still open for the case of positive characteristic.) The objects that we consider in this work are the exceptional simple Jordan pairs and triple systems, and a 56-dimensional exceptional simple structurable algebra referred to as the Brown algebra.
Gradings on these structures can be used to construct gradings on the simple exceptional Lie algebras of types E6 , E7 and E8, via the Kantor construction.
In this work, by grading we usually mean group grading. The base field is assumed to be algebraically closed of characteristic different from 2. Simple Lie algebras over the field of complex numbers C were classified
by W. Killing and E. Cartan in 1894, and Cartan decompositions (i.e., gradings by the associated root system) were used in the proof. After much work on some particular gradings, a systematic study of gradings was initiated by Patera and Zassenhaus in the context of Lie algebras, as a generalization of Cartan decompositions. Since then, gradings have been studied by many authors, and are an important tool for a better understanding of algebras and other structures. It is well-known that, if the base field is algebraically closed of characteristic 0, fine gradings by abelian groups are in bijective correspondence with maximal abelian diagonalizable subgroups of the automorphism group and, in that correspondence, equivalence classes of gradings correspond to conjugation classes of subgroups. Note that the same correspondence holds for any characteristic if we consider the automorphism group-scheme instead of the automorphism group. Therefore, the problem of classification of fine gradings by abelian groups on an algebra (and other algebraic structures) can be reformulated as an equivalent problem in group theory.
In Chapter 1 we recall the basic definitions of the algebraic structures that we study in this work. We also recall some well-known results of classifications of fine gradings on Cayley and Albert algebras, which constitute an extremely important tool to construct gradings in other structures, as shown in further sections. This chapter has no original results of the author. The basic definitions of gradings (by abelian groups) on Jordan systems are given in Chapter 2, and hereinafter some general results are proven. We recall the definition of the exceptional simple Jordan pairs and triple systems, i.e., the bi-Cayley and Albert pairs and triple systems. Some results related to the orbits and automorphism groups of the bi-Cayley pair and triple system are given too. In Chapters 3 and 4 we obtain classifications of the equivalence classes of fine gradings (by abelian groups) on the exceptional simple Jordan pairs and triple systems (over an algebraically closed field of characteristic not 2). The associated Weyl group is determined for each grading. We also study the induced fine gradings on e6 and e7 via the TKK-construction. Finally, in Chapter 5 we give a construction of a Z^3_4-grading on the Brown algebra. We then compute the Weyl group of this grading and study how this grading can be used to construct some very special fine gradings on the Lie algebras of types E6, E7 and E8.
Resumen (otro idioma):
Gradings on these structures can be used to construct gradings on the simple exceptional Lie algebras of types E6 , E7 and E8, via the Kantor construction.
In this work, by grading we usually mean group grading. The base field is assumed to be algebraically closed of characteristic different from 2. Simple Lie algebras over the field of complex numbers C were classified
by W. Killing and E. Cartan in 1894, and Cartan decompositions (i.e., gradings by the associated root system) were used in the proof. After much work on some particular gradings, a systematic study of gradings was initiated by Patera and Zassenhaus in the context of Lie algebras, as a generalization of Cartan decompositions. Since then, gradings have been studied by many authors, and are an important tool for a better understanding of algebras and other structures. It is well-known that, if the base field is algebraically closed of characteristic 0, fine gradings by abelian groups are in bijective correspondence with maximal abelian diagonalizable subgroups of the automorphism group and, in that correspondence, equivalence classes of gradings correspond to conjugation classes of subgroups. Note that the same correspondence holds for any characteristic if we consider the automorphism group-scheme instead of the automorphism group. Therefore, the problem of classification of fine gradings by abelian groups on an algebra (and other algebraic structures) can be reformulated as an equivalent problem in group theory.
In Chapter 1 we recall the basic definitions of the algebraic structures that we study in this work. We also recall some well-known results of classifications of fine gradings on Cayley and Albert algebras, which constitute an extremely important tool to construct gradings in other structures, as shown in further sections. This chapter has no original results of the author. The basic definitions of gradings (by abelian groups) on Jordan systems are given in Chapter 2, and hereinafter some general results are proven. We recall the definition of the exceptional simple Jordan pairs and triple systems, i.e., the bi-Cayley and Albert pairs and triple systems. Some results related to the orbits and automorphism groups of the bi-Cayley pair and triple system are given too. In Chapters 3 and 4 we obtain classifications of the equivalence classes of fine gradings (by abelian groups) on the exceptional simple Jordan pairs and triple systems (over an algebraically closed field of characteristic not 2). The associated Weyl group is determined for each grading. We also study the induced fine gradings on e6 and e7 via the TKK-construction. Finally, in Chapter 5 we give a construction of a Z^3_4-grading on the Brown algebra. We then compute the Weyl group of this grading and study how this grading can be used to construct some very special fine gradings on the Lie algebras of types E6, E7 and E8.
Resumen (otro idioma):
Pal. clave: álgebras no asociativas ; non-associative algebras
Área de conocimiento: Álgebra
Departamento: Matemáticas
Nota: Presentado: 02 02 2017
Nota: Tesis-Univ. Zaragoza, Matemáticas, 2017
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